3.2379 \(\int \frac {1}{(a+b \sqrt [3]{x})^3 x^3} \, dx\)

Optimal. Leaf size=146 \[ -\frac {84 b^6 \log \left (a+b \sqrt [3]{x}\right )}{a^9}+\frac {28 b^6 \log (x)}{a^9}+\frac {21 b^6}{a^8 \left (a+b \sqrt [3]{x}\right )}+\frac {63 b^5}{a^8 \sqrt [3]{x}}+\frac {3 b^6}{2 a^7 \left (a+b \sqrt [3]{x}\right )^2}-\frac {45 b^4}{2 a^7 x^{2/3}}+\frac {10 b^3}{a^6 x}-\frac {9 b^2}{2 a^5 x^{4/3}}+\frac {9 b}{5 a^4 x^{5/3}}-\frac {1}{2 a^3 x^2} \]

[Out]

3/2*b^6/a^7/(a+b*x^(1/3))^2+21*b^6/a^8/(a+b*x^(1/3))-1/2/a^3/x^2+9/5*b/a^4/x^(5/3)-9/2*b^2/a^5/x^(4/3)+10*b^3/
a^6/x-45/2*b^4/a^7/x^(2/3)+63*b^5/a^8/x^(1/3)-84*b^6*ln(a+b*x^(1/3))/a^9+28*b^6*ln(x)/a^9

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Rubi [A]  time = 0.10, antiderivative size = 146, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {266, 44} \[ -\frac {45 b^4}{2 a^7 x^{2/3}}-\frac {9 b^2}{2 a^5 x^{4/3}}+\frac {21 b^6}{a^8 \left (a+b \sqrt [3]{x}\right )}+\frac {3 b^6}{2 a^7 \left (a+b \sqrt [3]{x}\right )^2}+\frac {63 b^5}{a^8 \sqrt [3]{x}}+\frac {10 b^3}{a^6 x}-\frac {84 b^6 \log \left (a+b \sqrt [3]{x}\right )}{a^9}+\frac {28 b^6 \log (x)}{a^9}+\frac {9 b}{5 a^4 x^{5/3}}-\frac {1}{2 a^3 x^2} \]

Antiderivative was successfully verified.

[In]

Int[1/((a + b*x^(1/3))^3*x^3),x]

[Out]

(3*b^6)/(2*a^7*(a + b*x^(1/3))^2) + (21*b^6)/(a^8*(a + b*x^(1/3))) - 1/(2*a^3*x^2) + (9*b)/(5*a^4*x^(5/3)) - (
9*b^2)/(2*a^5*x^(4/3)) + (10*b^3)/(a^6*x) - (45*b^4)/(2*a^7*x^(2/3)) + (63*b^5)/(a^8*x^(1/3)) - (84*b^6*Log[a
+ b*x^(1/3)])/a^9 + (28*b^6*Log[x])/a^9

Rule 44

Int[((a_) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*
x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && L
tQ[m + n + 2, 0])

Rule 266

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rubi steps

\begin {align*} \int \frac {1}{\left (a+b \sqrt [3]{x}\right )^3 x^3} \, dx &=3 \operatorname {Subst}\left (\int \frac {1}{x^7 (a+b x)^3} \, dx,x,\sqrt [3]{x}\right )\\ &=3 \operatorname {Subst}\left (\int \left (\frac {1}{a^3 x^7}-\frac {3 b}{a^4 x^6}+\frac {6 b^2}{a^5 x^5}-\frac {10 b^3}{a^6 x^4}+\frac {15 b^4}{a^7 x^3}-\frac {21 b^5}{a^8 x^2}+\frac {28 b^6}{a^9 x}-\frac {b^7}{a^7 (a+b x)^3}-\frac {7 b^7}{a^8 (a+b x)^2}-\frac {28 b^7}{a^9 (a+b x)}\right ) \, dx,x,\sqrt [3]{x}\right )\\ &=\frac {3 b^6}{2 a^7 \left (a+b \sqrt [3]{x}\right )^2}+\frac {21 b^6}{a^8 \left (a+b \sqrt [3]{x}\right )}-\frac {1}{2 a^3 x^2}+\frac {9 b}{5 a^4 x^{5/3}}-\frac {9 b^2}{2 a^5 x^{4/3}}+\frac {10 b^3}{a^6 x}-\frac {45 b^4}{2 a^7 x^{2/3}}+\frac {63 b^5}{a^8 \sqrt [3]{x}}-\frac {84 b^6 \log \left (a+b \sqrt [3]{x}\right )}{a^9}+\frac {28 b^6 \log (x)}{a^9}\\ \end {align*}

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Mathematica [A]  time = 0.18, size = 130, normalized size = 0.89 \[ \frac {\frac {a \left (-5 a^7+8 a^6 b \sqrt [3]{x}-14 a^5 b^2 x^{2/3}+28 a^4 b^3 x-70 a^3 b^4 x^{4/3}+280 a^2 b^5 x^{5/3}+1260 a b^6 x^2+840 b^7 x^{7/3}\right )}{x^2 \left (a+b \sqrt [3]{x}\right )^2}-840 b^6 \log \left (a+b \sqrt [3]{x}\right )+280 b^6 \log (x)}{10 a^9} \]

Antiderivative was successfully verified.

[In]

Integrate[1/((a + b*x^(1/3))^3*x^3),x]

[Out]

((a*(-5*a^7 + 8*a^6*b*x^(1/3) - 14*a^5*b^2*x^(2/3) + 28*a^4*b^3*x - 70*a^3*b^4*x^(4/3) + 280*a^2*b^5*x^(5/3) +
1260*a*b^6*x^2 + 840*b^7*x^(7/3)))/((a + b*x^(1/3))^2*x^2) - 840*b^6*Log[a + b*x^(1/3)] + 280*b^6*Log[x])/(10
*a^9)

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fricas [A]  time = 0.51, size = 230, normalized size = 1.58 \[ \frac {280 \, a^{3} b^{9} x^{3} + 420 \, a^{6} b^{6} x^{2} + 90 \, a^{9} b^{3} x - 5 \, a^{12} - 840 \, {\left (b^{12} x^{4} + 2 \, a^{3} b^{9} x^{3} + a^{6} b^{6} x^{2}\right )} \log \left (b x^{\frac {1}{3}} + a\right ) + 840 \, {\left (b^{12} x^{4} + 2 \, a^{3} b^{9} x^{3} + a^{6} b^{6} x^{2}\right )} \log \left (x^{\frac {1}{3}}\right ) + 15 \, {\left (56 \, a b^{11} x^{3} + 98 \, a^{4} b^{8} x^{2} + 36 \, a^{7} b^{5} x - 3 \, a^{10} b^{2}\right )} x^{\frac {2}{3}} - 3 \, {\left (140 \, a^{2} b^{10} x^{3} + 224 \, a^{5} b^{7} x^{2} + 63 \, a^{8} b^{4} x - 6 \, a^{11} b\right )} x^{\frac {1}{3}}}{10 \, {\left (a^{9} b^{6} x^{4} + 2 \, a^{12} b^{3} x^{3} + a^{15} x^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b*x^(1/3))^3/x^3,x, algorithm="fricas")

[Out]

1/10*(280*a^3*b^9*x^3 + 420*a^6*b^6*x^2 + 90*a^9*b^3*x - 5*a^12 - 840*(b^12*x^4 + 2*a^3*b^9*x^3 + a^6*b^6*x^2)
*log(b*x^(1/3) + a) + 840*(b^12*x^4 + 2*a^3*b^9*x^3 + a^6*b^6*x^2)*log(x^(1/3)) + 15*(56*a*b^11*x^3 + 98*a^4*b
^8*x^2 + 36*a^7*b^5*x - 3*a^10*b^2)*x^(2/3) - 3*(140*a^2*b^10*x^3 + 224*a^5*b^7*x^2 + 63*a^8*b^4*x - 6*a^11*b)
*x^(1/3))/(a^9*b^6*x^4 + 2*a^12*b^3*x^3 + a^15*x^2)

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giac [A]  time = 0.17, size = 123, normalized size = 0.84 \[ -\frac {84 \, b^{6} \log \left ({\left | b x^{\frac {1}{3}} + a \right |}\right )}{a^{9}} + \frac {28 \, b^{6} \log \left ({\left | x \right |}\right )}{a^{9}} + \frac {840 \, a b^{7} x^{\frac {7}{3}} + 1260 \, a^{2} b^{6} x^{2} + 280 \, a^{3} b^{5} x^{\frac {5}{3}} - 70 \, a^{4} b^{4} x^{\frac {4}{3}} + 28 \, a^{5} b^{3} x - 14 \, a^{6} b^{2} x^{\frac {2}{3}} + 8 \, a^{7} b x^{\frac {1}{3}} - 5 \, a^{8}}{10 \, {\left (b x^{\frac {1}{3}} + a\right )}^{2} a^{9} x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b*x^(1/3))^3/x^3,x, algorithm="giac")

[Out]

-84*b^6*log(abs(b*x^(1/3) + a))/a^9 + 28*b^6*log(abs(x))/a^9 + 1/10*(840*a*b^7*x^(7/3) + 1260*a^2*b^6*x^2 + 28
0*a^3*b^5*x^(5/3) - 70*a^4*b^4*x^(4/3) + 28*a^5*b^3*x - 14*a^6*b^2*x^(2/3) + 8*a^7*b*x^(1/3) - 5*a^8)/((b*x^(1
/3) + a)^2*a^9*x^2)

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maple [A]  time = 0.00, size = 123, normalized size = 0.84 \[ \frac {3 b^{6}}{2 \left (b \,x^{\frac {1}{3}}+a \right )^{2} a^{7}}+\frac {21 b^{6}}{\left (b \,x^{\frac {1}{3}}+a \right ) a^{8}}+\frac {28 b^{6} \ln \relax (x )}{a^{9}}-\frac {84 b^{6} \ln \left (b \,x^{\frac {1}{3}}+a \right )}{a^{9}}+\frac {63 b^{5}}{a^{8} x^{\frac {1}{3}}}-\frac {45 b^{4}}{2 a^{7} x^{\frac {2}{3}}}+\frac {10 b^{3}}{a^{6} x}-\frac {9 b^{2}}{2 a^{5} x^{\frac {4}{3}}}+\frac {9 b}{5 a^{4} x^{\frac {5}{3}}}-\frac {1}{2 a^{3} x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(b*x^(1/3)+a)^3/x^3,x)

[Out]

3/2*b^6/a^7/(b*x^(1/3)+a)^2+21*b^6/a^8/(b*x^(1/3)+a)-1/2/a^3/x^2+9/5*b/a^4/x^(5/3)-9/2*b^2/a^5/x^(4/3)+10*b^3/
a^6/x-45/2*b^4/a^7/x^(2/3)+63*b^5/a^8/x^(1/3)-84*b^6*ln(b*x^(1/3)+a)/a^9+28/a^9*b^6*ln(x)

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maxima [A]  time = 0.53, size = 132, normalized size = 0.90 \[ \frac {840 \, b^{7} x^{\frac {7}{3}} + 1260 \, a b^{6} x^{2} + 280 \, a^{2} b^{5} x^{\frac {5}{3}} - 70 \, a^{3} b^{4} x^{\frac {4}{3}} + 28 \, a^{4} b^{3} x - 14 \, a^{5} b^{2} x^{\frac {2}{3}} + 8 \, a^{6} b x^{\frac {1}{3}} - 5 \, a^{7}}{10 \, {\left (a^{8} b^{2} x^{\frac {8}{3}} + 2 \, a^{9} b x^{\frac {7}{3}} + a^{10} x^{2}\right )}} - \frac {84 \, b^{6} \log \left (b x^{\frac {1}{3}} + a\right )}{a^{9}} + \frac {28 \, b^{6} \log \relax (x)}{a^{9}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b*x^(1/3))^3/x^3,x, algorithm="maxima")

[Out]

1/10*(840*b^7*x^(7/3) + 1260*a*b^6*x^2 + 280*a^2*b^5*x^(5/3) - 70*a^3*b^4*x^(4/3) + 28*a^4*b^3*x - 14*a^5*b^2*
x^(2/3) + 8*a^6*b*x^(1/3) - 5*a^7)/(a^8*b^2*x^(8/3) + 2*a^9*b*x^(7/3) + a^10*x^2) - 84*b^6*log(b*x^(1/3) + a)/
a^9 + 28*b^6*log(x)/a^9

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mupad [B]  time = 1.21, size = 125, normalized size = 0.86 \[ \frac {\frac {4\,b\,x^{1/3}}{5\,a^2}-\frac {1}{2\,a}+\frac {14\,b^3\,x}{5\,a^4}-\frac {7\,b^2\,x^{2/3}}{5\,a^3}+\frac {126\,b^6\,x^2}{a^7}-\frac {7\,b^4\,x^{4/3}}{a^5}+\frac {28\,b^5\,x^{5/3}}{a^6}+\frac {84\,b^7\,x^{7/3}}{a^8}}{a^2\,x^2+b^2\,x^{8/3}+2\,a\,b\,x^{7/3}}-\frac {168\,b^6\,\mathrm {atanh}\left (\frac {2\,b\,x^{1/3}}{a}+1\right )}{a^9} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^3*(a + b*x^(1/3))^3),x)

[Out]

((4*b*x^(1/3))/(5*a^2) - 1/(2*a) + (14*b^3*x)/(5*a^4) - (7*b^2*x^(2/3))/(5*a^3) + (126*b^6*x^2)/a^7 - (7*b^4*x
^(4/3))/a^5 + (28*b^5*x^(5/3))/a^6 + (84*b^7*x^(7/3))/a^8)/(a^2*x^2 + b^2*x^(8/3) + 2*a*b*x^(7/3)) - (168*b^6*
atanh((2*b*x^(1/3))/a + 1))/a^9

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sympy [A]  time = 11.46, size = 706, normalized size = 4.84 \[ \begin {cases} \frac {\tilde {\infty }}{x^{3}} & \text {for}\: a = 0 \wedge b = 0 \\- \frac {1}{2 a^{3} x^{2}} & \text {for}\: b = 0 \\- \frac {1}{3 b^{3} x^{3}} & \text {for}\: a = 0 \\- \frac {5 a^{8} x^{\frac {2}{3}}}{10 a^{11} x^{\frac {8}{3}} + 20 a^{10} b x^{3} + 10 a^{9} b^{2} x^{\frac {10}{3}}} + \frac {8 a^{7} b x}{10 a^{11} x^{\frac {8}{3}} + 20 a^{10} b x^{3} + 10 a^{9} b^{2} x^{\frac {10}{3}}} - \frac {14 a^{6} b^{2} x^{\frac {4}{3}}}{10 a^{11} x^{\frac {8}{3}} + 20 a^{10} b x^{3} + 10 a^{9} b^{2} x^{\frac {10}{3}}} + \frac {28 a^{5} b^{3} x^{\frac {5}{3}}}{10 a^{11} x^{\frac {8}{3}} + 20 a^{10} b x^{3} + 10 a^{9} b^{2} x^{\frac {10}{3}}} - \frac {70 a^{4} b^{4} x^{2}}{10 a^{11} x^{\frac {8}{3}} + 20 a^{10} b x^{3} + 10 a^{9} b^{2} x^{\frac {10}{3}}} + \frac {280 a^{3} b^{5} x^{\frac {7}{3}}}{10 a^{11} x^{\frac {8}{3}} + 20 a^{10} b x^{3} + 10 a^{9} b^{2} x^{\frac {10}{3}}} + \frac {280 a^{2} b^{6} x^{\frac {8}{3}} \log {\relax (x )}}{10 a^{11} x^{\frac {8}{3}} + 20 a^{10} b x^{3} + 10 a^{9} b^{2} x^{\frac {10}{3}}} - \frac {840 a^{2} b^{6} x^{\frac {8}{3}} \log {\left (\frac {a}{b} + \sqrt [3]{x} \right )}}{10 a^{11} x^{\frac {8}{3}} + 20 a^{10} b x^{3} + 10 a^{9} b^{2} x^{\frac {10}{3}}} + \frac {1260 a^{2} b^{6} x^{\frac {8}{3}}}{10 a^{11} x^{\frac {8}{3}} + 20 a^{10} b x^{3} + 10 a^{9} b^{2} x^{\frac {10}{3}}} + \frac {560 a b^{7} x^{3} \log {\relax (x )}}{10 a^{11} x^{\frac {8}{3}} + 20 a^{10} b x^{3} + 10 a^{9} b^{2} x^{\frac {10}{3}}} - \frac {1680 a b^{7} x^{3} \log {\left (\frac {a}{b} + \sqrt [3]{x} \right )}}{10 a^{11} x^{\frac {8}{3}} + 20 a^{10} b x^{3} + 10 a^{9} b^{2} x^{\frac {10}{3}}} + \frac {840 a b^{7} x^{3}}{10 a^{11} x^{\frac {8}{3}} + 20 a^{10} b x^{3} + 10 a^{9} b^{2} x^{\frac {10}{3}}} + \frac {280 b^{8} x^{\frac {10}{3}} \log {\relax (x )}}{10 a^{11} x^{\frac {8}{3}} + 20 a^{10} b x^{3} + 10 a^{9} b^{2} x^{\frac {10}{3}}} - \frac {840 b^{8} x^{\frac {10}{3}} \log {\left (\frac {a}{b} + \sqrt [3]{x} \right )}}{10 a^{11} x^{\frac {8}{3}} + 20 a^{10} b x^{3} + 10 a^{9} b^{2} x^{\frac {10}{3}}} & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b*x**(1/3))**3/x**3,x)

[Out]

Piecewise((zoo/x**3, Eq(a, 0) & Eq(b, 0)), (-1/(2*a**3*x**2), Eq(b, 0)), (-1/(3*b**3*x**3), Eq(a, 0)), (-5*a**
8*x**(2/3)/(10*a**11*x**(8/3) + 20*a**10*b*x**3 + 10*a**9*b**2*x**(10/3)) + 8*a**7*b*x/(10*a**11*x**(8/3) + 20
*a**10*b*x**3 + 10*a**9*b**2*x**(10/3)) - 14*a**6*b**2*x**(4/3)/(10*a**11*x**(8/3) + 20*a**10*b*x**3 + 10*a**9
*b**2*x**(10/3)) + 28*a**5*b**3*x**(5/3)/(10*a**11*x**(8/3) + 20*a**10*b*x**3 + 10*a**9*b**2*x**(10/3)) - 70*a
**4*b**4*x**2/(10*a**11*x**(8/3) + 20*a**10*b*x**3 + 10*a**9*b**2*x**(10/3)) + 280*a**3*b**5*x**(7/3)/(10*a**1
1*x**(8/3) + 20*a**10*b*x**3 + 10*a**9*b**2*x**(10/3)) + 280*a**2*b**6*x**(8/3)*log(x)/(10*a**11*x**(8/3) + 20
*a**10*b*x**3 + 10*a**9*b**2*x**(10/3)) - 840*a**2*b**6*x**(8/3)*log(a/b + x**(1/3))/(10*a**11*x**(8/3) + 20*a
**10*b*x**3 + 10*a**9*b**2*x**(10/3)) + 1260*a**2*b**6*x**(8/3)/(10*a**11*x**(8/3) + 20*a**10*b*x**3 + 10*a**9
*b**2*x**(10/3)) + 560*a*b**7*x**3*log(x)/(10*a**11*x**(8/3) + 20*a**10*b*x**3 + 10*a**9*b**2*x**(10/3)) - 168
0*a*b**7*x**3*log(a/b + x**(1/3))/(10*a**11*x**(8/3) + 20*a**10*b*x**3 + 10*a**9*b**2*x**(10/3)) + 840*a*b**7*
x**3/(10*a**11*x**(8/3) + 20*a**10*b*x**3 + 10*a**9*b**2*x**(10/3)) + 280*b**8*x**(10/3)*log(x)/(10*a**11*x**(
8/3) + 20*a**10*b*x**3 + 10*a**9*b**2*x**(10/3)) - 840*b**8*x**(10/3)*log(a/b + x**(1/3))/(10*a**11*x**(8/3) +
20*a**10*b*x**3 + 10*a**9*b**2*x**(10/3)), True))

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